The normal distribution

Measure the height of 1000 people and enter the values into a bar chart. What appears is not a jumble but a surprisingly even shape: high in the middle, falling away symmetrically on both sides. You get the same shape for measurement errors, shoe sizes, reaction times and exam scores. It is called the bell curve, and the distribution behind it is the normal distribution.

Why the same shape so often?

The reason is always the same: when many small, mutually independent influences come together, they usually partly cancel each other out. Extreme values only arise when nearly all the influences point the same way, and that is rare. Height depends on many genes and on nutrition, a measurement error on many small disturbances. That is why values pile up in the middle.

Two numbers fix everything

Every bell curve is completely determined by two numbers. The mean $\mu$ says where the peak sits. The standard deviation $\sigma$ says how wide the bell is. A large $\sigma$ gives a flat, broad curve, a small one a narrow, tall curve. The area under the curve always stays the same, namely 1, because some value is certain to occur.

The sigma rules

These three numbers are worth memorising, because they hold for every normal distribution, whatever $\mu$ and $\sigma$ are. About 68% of all values lie at most one standard deviation from the mean. About 95% lie within two, and about 99.7% within three. So anything beyond three standard deviations really is exceptional.

An example: the height of adult men in Germany is roughly $\mu = 180$ cm with $\sigma = 7$ cm. Then about 68% are between 173 cm and 187 cm tall, about 95% between 166 cm and 194 cm. Anyone above 201 cm belongs to the top 0.15%, because that is three standard deviations up.